theoretical model of dynamic spin polarization of nuclei...

Theoretical model of dynamic spin polarization

of nuclei coupled to paramagnetic point defects

in diamond and silicon carbide

Viktor Ivády, Krisztian Szasz, Abram L. Falk, Paul V. Klimov, David J. Christle, Erik

Janzén, Igor Abrikosov, David D. Awschalom and Adam Gali

Linköping University Post Print

N.B.: When citing this work, cite the original article.

Original Publication:

Viktor Ivády, Krisztian Szasz, Abram L. Falk, Paul V. Klimov, David J. Christle, Erik Janzén,

Igor Abrikosov, David D. Awschalom and Adam Gali, Theoretical model of dynamic spin

polarization of nuclei coupled to paramagnetic point defects in diamond and silicon carbide,

2015, Physical Review B. Condensed Matter and Materials Physics, (92), 11, 115206.

http://dx.doi.org/10.1103/PhysRevB.92.115206

Copyright: American Physical Society

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Postprint available at: Linköping University Electronic Press

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PHYSICAL REVIEW B 92, 115206 (2015)

Theoretical model of dynamic spin polarization of nuclei coupled to paramagneticpoint defects in diamond and silicon carbide

Viktor Ivady,1,2 Krisztian Szasz,2 Abram L. Falk,3,4 Paul V. Klimov,3,5 David J. Christle,3,5 Erik Janzen,1

Igor A. Abrikosov,1,6,7 David D. Awschalom,3 and Adam Gali2,8,*

1Department of Physics, Chemistry and Biology, Linkoping University, SE-581 83 Linkoping, Sweden2Wigner Research Centre for Physics, Hungarian Academy of Sciences, PO Box 49, H-1525, Budapest, Hungary

3Institute for Molecular Engineering, University of Chicago, Chicago, Illinois, USA4IBM T.J. Watson Research Center, Yorktown Heights, NY, USA

5Department of Physics, University of California, Santa Barbara, California, USA6Materials Modeling and Development Laboratory, National University of Science and Technology “MISIS,” 119049 Moscow, Russia

7LACOMAS Laboratory, Tomsk State University, 634050 Tomsk, Russia8Department of Atomic Physics, Budapest University of Technology and Economics, Budafoki ut 8., H-1111 Budapest, Hungary

(Received 26 May 2015; revised manuscript received 1 September 2015; published 18 September 2015)

Dynamic nuclear spin polarization (DNP) mediated by paramagnetic point defects in semiconductors is akey resource for both initializing nuclear quantum memories and producing nuclear hyperpolarization. DNPis therefore an important process in the field of quantum-information processing, sensitivity-enhanced nuclearmagnetic resonance, and nuclear-spin-based spintronics. DNP based on optical pumping of point defects hasbeen demonstrated by using the electron spin of nitrogen-vacancy (NV) center in diamond, and more recently, byusing divacancy and related defect spins in hexagonal silicon carbide (SiC). Here, we describe a general modelfor these optical DNP processes that allows the effects of many microscopic processes to be integrated. Applyingthis theory, we gain a deeper insight into dynamic nuclear spin polarization and the physics of diamond and SiCdefects. Our results are in good agreement with experimental observations and provide a detailed and unifiedunderstanding. In particular, our findings show that the defect electron spin coherence times and excited statelifetimes are crucial factors in the entire DNP process.

DOI: 10.1103/PhysRevB.92.115206 PACS number(s): 76.30.Mi, 71.15.Mb, 76.70.Hb, 61.72.jn

I. INTRODUCTION

Point defects in solids are promising implementations ofquantum bits for quantum computing [1,2]. In particular,the negatively charged nitrogen-vacancy defect (NV center)in diamond [3] has become a leading system in solid-state quantum-information processing because of its uniquemagnetooptical properties, including long spin coherencetimes [4] and the ease of optical initialization and readoutof its spin state [5], even nondestructively [6,7]. Since ithas a high-spin electronic structure similar to the NV centerin diamond, the divacancy in silicon carbide (SiC) has alsobeen proposed to serve as a solid-state quantum bit [8]. Infact, SiC hosts many other color centers that may also actas quantum bits [8–13]. Recent demonstrations have showncoherent manipulation of divacancy and related defect spinsin 4H- [14], 6H- [15,16], and 3C-SiC [15]. Coherent controlof the electronic spin of the negatively charged Si vacancyhas also been investigated [13,17,18]. Further milestoneson the path towards robust SiC-based quantum-informationtechnology have been the findings that isolated divacancyqubits have ∼1 ms at low temperatures [19] and that isolatedSi-vacancy qubits can operate at room temperature [20].

Coherent control of the electron spins of paramagneticpoint defects makes it possible to control and manipulateother spins in the vicinity of the point defect. For in-stance, the proximate nuclear spins of the NV center indiamond can be polarized [21–28], which can be a basis for

*[email protected]

quantum memories [29–32], entanglement-based metrologicaldevices [33], and solid-state nuclear gyroscopes [34,35]. Arecent demonstration has shown that nuclear spins proximateto divacancies and related defects in 4H- and 6H-SiC canbe effectively polarized [36], an important step towardsenabling long-lived quantum-information processing in thistechnologically mature semiconductor material. The transferof the point defects’ electron spin polarization can also leadto hyperpolarization of the host material, thereby enablingsensitivity-enhanced nuclear magnetic resonance and spin-tronic applications [37–40].

Many of these applications rely on dynamic nuclear spinpolarization (DNP) to mediate polarization transfer fromthe electron spin to neighboring nuclear spins through thehyperfine interaction. Therefore a fundamental understandingof DNP processes is an important aspect of the technologicaldevelopment of nuclear spintronics.

Jacques et al. [23] developed an insightful spin Hamiltonianmodel that describes the polarization process of 15N nucleiat the avoided crossing of the diamond NV centers spinsublevels, otherwise known as the level anticrossing (LAC).The transverse part of the hyperfine interaction is responsiblefor the exchange of electron and nuclear polarization [25],while the spin selective nonradiative decay of the electron fromthe excited state (ES) is responsible for the maintenance of theelectron-spin polarization. Continuous cycling of optical elec-tronic excitation, flip-flops of electronic and nuclear spins, andnonradiative decay result in a polarization of both the electronand the nuclear spin populations. The so-called “excited-statelevel anticrossing (ESLAC) mechanism” is when spin flip flopspredominantly occur in the excited state. At larger magnetic

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VIKTOR IVADY et al. PHYSICAL REVIEW B 92, 115206 (2015)

fields, when the LAC occurs in the ground state (GS), theanalogous ground-state level anticrossing (GSLAC) processoccurs [21,39]. To understand the particular features observedin the GS dynamic nuclear spin polarization of a 13C nucleiadjacent to the vacancy of the NV center, Wang et al. [39]recently developed a model capable of describing generalhyperfine interactions, i.e., for nuclei not on the symmetryaxis. Furthermore, to take into account the effect of externalstrain and spin relaxation and dephasing processes, Fischeret al. [28] recently proposed a density-matrix model.

While these models capture certain phenomena of DNP, amore general model showing predictive power over severalcolor-center systems would be an important development.Here, we propose an extended model that (i) handles generalhyperfine interactions of paramagnetic defects with symmet-rically or nonsymmetrically placed nuclei spins, (ii) takesinto account simultaneous GSLAC and ESLAC processes,and (iii) tracks the evolution of spins with time explicitlyparameterized. Our model provides insight into the phenomenaof the dynamic nuclear spin polarization mechanism forboth the NV center in diamond and the divacancies in SiC,and explains several experimental observations. Throughoutour investigation, electron-spin decoherence appears as animportant limiting factor for the polarizability of the nuclearspins. We show that considering electron-spin coherence willbe vital to maximizing the performance of DNP in practicalapplications.

In Sec. II, we briefly describe the electronic structure andthe corresponding spin properties of the ground and excitedstates of the considered point defects, namely, the NV center indiamond and the divacancy defects in 4H- and 6H-SiC. Thesedefects’ structures are then used to define the spin Hamiltonian,which is given together with the model of the dynamic nuclearspin polarization in Sec. III. In Sec. III, we also summarizethe basic concept and parameters of the model to calculate thenuclear spin polarization as a function of different variables.In Sec. IV, we describe the ab initio methods and modelsthat we use to calculate the spin related properties of NVcenter in diamond and divacancy defects in SiC, along withthe corresponding results. The full hyperfine tensors of thestudied nuclear spins are calculated both in the ground andexcited states and are important parameters in the DNP model.In Sec. V, we provide the results and an analysis of DNP forthe NV center in diamond and the divacancy in SiC. Finally,we summarize our findings in Sec. VI.

II. ELECTRONIC STRUCTURE OF NV CENTER INDIAMOND AND DIVACANCY IN SIC

The geometry and the electronic structure of the negativelycharged NV center in diamond have been discussed previ-ously based on highly convergent ab initio plane-wave largesupercell calculations [25,41,42]. The diamond NV center isa complex that consists of a substitutional nitrogen adjacentto a vacancy in diamond and possesses C3v symmetry (seeFig. 1). The defect exhibits a fully occupied lower a1 level, anda double-degenerate upper e level filled by two parallel-spinelectrons in the gap with an S = 1 high-spin ground state.The S = 1 excited state is well described by the promotionof an electron from the lower defect level to the upper level

-1 | (d) (e)

MS = 0

MS = ±1

MS = ±1 MS = 0

3A2

1A1

3E

1E

ES: 3E

GS: 3A2

e a1

Valence band

(b)

Conduction band

0 | 0 |

0 | -1 0 | |

A

(a) C3

C

NVC

B = 0

BLAC

DES

DGS

D

(c)

B

Ener

gy

DGS

DES

GSLAC

ESLAC

MS = 0

MS = +1

MS = 0

MS = ±1

FIG. 1. (Color online) A schematic diagram of (a) the structureshowing the C3 rotation axis and (b) the electron configuration ofthe NV center in diamond. (c) The magnetic field dependence ofthe ES and GS spin-sublevel energies, showing the ESLAC andGSLAC. (d) The NV center’s optical polarization cycle and (e) thenuclear spin polarization cycle. In (c) and (d), DGS = 2.87 GHz,DES = 1.42 GHz are the zero-field constants. In (d), the green dashedarrows represent the nonradiative decays, which are mediated byspin-orbit couplings and vibrations. The thick grey arrows representthe optical absorption/emission paths. The wavy lines representphoton absorption/emission in the visible (red) and near-infrared(brown) regions. (e) At zero magnetic field (B = 0), the |0↓〉 level isseparated by the zero-field constant from the |−1↑〉 level. Applying aB = BLAC > 0 field causes the two states to form an avoided crossing,where the small gap is introduced by the hyperfine interaction (A).In this condition, hyperfine coupling (brown circular arrow) andnonradiative decay (green dotted line) are responsible for the nuclearspin polarization in the optical cycle.

in the gap [43]. The electron spin may interact with nuclearspins: 14N or 15N possessing I = 1 or I = 1/2 nuclear spin,respectively, or 13C with I = 1/2.

The hyperfine interaction between the electron spin andnuclear spin has been studied by means of ab initio methods inprevious publications, both in the GS [26,41,42,44] and in theES [25]. In the GS, the electronic spin-spin dipole interactioncauses the fine electron structure to have a zero-field splitting,where “zero field” refers to zero external magnetic field. Inhigh-purity and low-strain diamond samples, this splitting canbe described by a single parameter, DGS = 2.87 GHz, whichseparates the mS = 0 and the mS = ±1 sublevels within theS = 1 manifold. At room temperature, the fine structure inthe electronic ES shows a similar feature, except that its zero-field splitting is only DES = 1.42 GHz. At lower temperatures,the fine structure becomes more complicated, hindering off-resonantly pumped DNP in the ES [45]. Between the ES andGS triplets, nonradiative relaxation pathways through singletstates selectively flip mS = ±1 states to mS = 0 state in theoptical excitation cycle [46–50], which allows optical spin-polarization of the NV center (Fig. 1).

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THEORETICAL MODEL OF DYNAMIC SPIN . . . PHYSICAL REVIEW B 92, 115206 (2015)

ES: 3E

GS: 3A2

e a1

Valence band

(b)

Conduction band

(a) C3

C

VSi

e

VC Si

= SiIIb = SiIIa

FIG. 2. (Color online) A schematic diagram of the structure ofthe divacancy in SiC. The defect levels in the gap as well as thecorresponding ground and excited states are shown. The high-energyupper empty e level does not play a role in the excitation process.Those Si atoms that participate in DNP are labeled by SiIIa and SiIIb.The similarity between the electronic structure of the NV center indiamond (cf. Fig. 1) and that of the SiC divacancy is apparent.

We also study the divacancy defects in SiC. Before describ-ing the electronic structure of the divacancy, we will brieflydiscuss the host semiconductor. SiC has about 250 knowndifferent polytypes, which share the same basal hexagonallattice but have different stacking sequences of Si-C bilayersperpendicular to this plane. The most industrially importantpolytypes are the 4H and 6H polytypes. The inequivalencyof the crystal planes leads to the so-called h and k types ofbilayers in 4H-SiC, and h, k1, k2 types of bilayers in 6H-SiC.In the inequivalent bilayers, the crystalline environment in thesecond, third, etc., neighborhood will be different, yieldinga similar but quantitatively distinguishable electron structurefor the corresponding point defects. For divacancy defects,adjacent silicon and carbon atoms are absent. In 4H-SiC, thefour inequivalent forms of divacancy are the hh, kk, hk, andkh configurations. The first two (axial) configurations haveC3v symmetry, and the second two (basal-plane oriented)configurations have C1h symmetry. We focus our study hereon the axial configurations, whose C3v symmetry is the sameas that of the NV center in diamond. In 6H-SiC, there are threeaxial configurations of divacancies: the hh, k1k1, and k2k2 con-figurations. In 4H-SiC, the hh and kk configurations have beenalready associated with zero-phonon-line photoluminescencepeaks and spin transitions, measured with optically detectedmagnetic resonance (ODMR) signals and sometimes called thePL1 and PL2 centers [14,51,52]. An additional center has beenfound, PL6, which also exhibits ODMR and similar physicalproperties to the axial divacancies [14,36,51]. The physicalstructure of PL6 has not yet been identified. In 6H-SiC,the PL and ODMR lines that have been labeled QL1, QL2,and QL6 [15] are also associated with the axial divacancies(Sec. IV), namely, the k1k1, hh, and k2k2 configurations,respectively. All these axial divacancies share the same elec-tronic structure depicted in Fig. 2. The carbon dangling bondsof the Si-vacancy part of the defect create a double-degeneratee-level close the valence band edge, which is occupied bytwo electrons with parallel spins. Thus, the neutral divacancyhas a high spin (S = 1) ground state. The excited state may bedescribed as the promotion of an electron from the lower defecta1 level to this e level [8,12], akin to that of the NV center indiamond. In SiC crystals, beside 13C isotopes (with a 1.3% nat-

ural abundance), 29Si isotopes (with a 4.7% natural abundance)have I = 1/2 nuclear spins that may interact with the divacan-cies’ S = 1 electron spins. Very little is known about the natureof the triplet ES and the dark singlet states responsible forthe electron spin polarization for divacancy in SiC. However,recent measurements indicate [36] that the triplet ES has asimilar electronic structure to that of NV center in diamond.Experimental results imply C3v symmetry in the ES of axialdivacancies even at low temperatures [36]. In this work, we as-sume that the SiC divacancy has a similar model for the opticalspin polarization processes as does the NV center in diamond.The measured DGS and DES zero-field splittings of axial diva-cancies in 4H and 6H polytypes are summarized in Table II.

III. MODEL OF THE DYNAMIC NUCLEAR SPINPOLARIZATION PROCESS

A. Modeling the dynamic nuclear spin polarizationin the optical cycle

The degree of nuclear spin polarization (P ) is the main ob-servable in DNP measurements. Understanding its dependenceon external variables like magnetic field and temperature,and internal variables like the details of the hyperfine tensor,has great importance from an applications point of view. Atheoretical model is a crucial component of this understanding.

In the forthcoming section, we briefly review the model ofJacques et al. [23] on the dynamic nuclear spin polarization ofthe NV center’s nitrogen nucleus and extend it by a generalizedderivation of the basic equations. Next, we describe ourmodel that takes into account many microscopic featuresand processes that have been overlooked or not integrated inprevious models. The most important features that we integrateare the electron spin decoherence in the ground and excitedstate, the short lifetime of the excited state, the anisotropy ofthe hyperfine tensors, the angle of the external magnetic field,and the overlap of the ground and excited state’s spin flippingprocesses.

1. Previous model of dynamic nuclear spin polarization

The dynamic nuclear spin polarization processes are sen-sitive to the details of the spin Hamiltonian of the consideredpoint defect and the optical electron spin polarization cycle.In the simplest case, e.g., an NV center in diamond withan adjacent I = 1/2 nuclear spin located on the symmetryaxis of the defect, the process can be understood as atwo-step mechanism. Continuous optical excitation polarizesthe electron spin in the MS = 0 spin state, while the nuclearspin may be any linear combination of the spin-up |↑〉 andspin-down |↓〉 states. Near the vicinity of a LAC, the hyperfineinteraction couples the nuclear and electron spins effectivelyand rotates the two spin state |0↓〉 into the state |−1↑〉. Thisstate is then transformed into the state |0↑〉 by the opticalexcitation and decay processes shown in Fig. 1(e). As aresult, the nuclear spin component |↓〉 is flipped and |↑〉 ispredominantly populated.

The spin Hamiltonian that governs the nuclear spin flipsis well known for the NV center in diamond [23]. For asingle nuclear spin adjacent to the defect, the ground-state

115206-3


Hamiltonian of the electron-nuclear spin system can be writtenas

HGS = STDGSS + μBBTgeS + STAGSI + μNBTgNI, (1)

where S and I are the electron and nuclear spin operators, DGS

and AGS are the tensors of zero-field and hyperfine interactionin the ground-state electron spin configuration of the defect,respectively, B is the external magnetic field, ge and gN arethe g tensors of the electron and nuclear spin, and μB andμN are the Bohr and nuclear magnetons, respectively. Forthe considered defect, the g tensors are nearly isotropic andsimplify to a scalar value, such as ge = 2.0023.

For the NV center, the zero-field interaction term Hzfs, thefirst term on left-hand side of Eq. (1), can be written as

Hzfs = STDGSS = DGS(S2

z − 23

). (2)

The hyperfine interaction term Hhyp, which is the third termon the left-hand side of Eq. (1), couples the electron and nuclearspins and therefore plays a key role in DNP. When the nuclearspin resides on the symmetry axis of the defect the hyperfinetensor A is diagonal with diagonal elements A⊥ and A‖. Thehyperfine interaction term is written as [25]

Hhyp = STAGSI = AGS⊥

S+I− + S−I+2

+ AGS‖ SzIz, (3)

where S± and I± are the electron and nuclear spin ladderoperators, respectively, and Sz and Iz are the z components ofthe electron and nuclear spins, respectively. The first term onthe left-hand side of Eq. (3) is responsible for the flipping ofthe nuclear and electron spins governed by A⊥.

The NV center in diamond possesses C3v symmetry in itsexcited state (ES) at elevated temperatures [41], and thereforethe excited state Hamiltonian has the same form as Eq. (1).However, the zero-field-splitting tensor DES and the hyperfinetensor AES differ from those in the ground-state electronicconfiguration.

For the sake of a general description, we utilize the densitymatrix formalism[53] to derive the steady-state nuclear spinpolarization of the dynamical process. First, we express thewave function of an electron and nuclear spin system in abasis

� =∑m,n

Cmn(t)|mn〉, (4)

where |mn〉 = |m〉 ⊗ |n〉, m and n are the electron andnuclear spin projections on the quantization axis and Cmn arecoefficients. The spin density matrix of the system can beobtained from the coefficients,

ρmn,m′n′ (t) = Cmn(t)C∗m′n′(t), (5)

while the nuclear spin density matrix can be obtained from thepartial trace of ρmn,m′n′ ,

�nn′ (t) =∑m

ρmn,mn′ (t). (6)

To describe the time evolution of the diagonal elements of thenuclear spin density matrix in the dynamical process, for time

scales larger than the average length of an optical cycle, wewrite the kinetic equations in the form

�nn = c(J )��nn − η�nn, (7)

where the dot represents time differentiation. The second termon the right-hand side describes the nuclear spin relaxation dueto the environment. Note that all the nuclear spin projectionsare assumed to relax equally with rate η. The first term onthe right-hand side describes the change of the nuclear spinprojection due to the interaction with the electron spin and theexternal magnetic field. c(J ) is the number of optical cycle perunit time and ��nn is the averaged variation of the nuclearspin projection during the free evolution time, i.e., between twooptical excitations. This last term can be determined from thespin Hamiltonian of the system. Note that both terms dependon the intensity J of the excitation laser. At low intensities, therate of optical excitation c(J ) depends linearly on the intensity.For strong laser excitation, when the rate of excitation may becomparable with the periodicity of the spin rotations due tohyperfine interaction or transverse magnetic field, the timeaverage of ��nn depends on J too. In the following, weassume weak laser intensities, and thus only c(J ) depends onthe intensity. Generally, ��nn can be written as

��nn =∑n′

pn′n�n′n′ − �nn

∑n′

pnn′ , (8)

where pnn′ is the average probability of flipping the nuclearspin from |n〉 to |n′〉. The first and second terms on theright-hand side describe the probabilities of flipping the spinin and out of the state |n〉, respectively. We note that for theHamiltonian specified in Eqs. (1)–(3), n′ = n ± 1 holds [54].In the following, we restrict our derivation to the case ofI = 1/2, however, we do not impose any other losses in thegenerality of the spin Hamiltonian. In this case, Eq. (8) readsas

��± 12 ± 1

2= p±�∓ 1

2 ∓ 12− p∓�± 1

2 ± 12, (9)

where p± are the average probabilities of raising and loweringof the nuclear spin projection between two optical cycles.

When dynamic nuclear spin polarization is in a stationarystate, the different spin rotation processes are balanced and

�+ 12 + 1

2= �− 1

2 − 12. (10)

By using the definition P = �+ 12 + 1

2− �− 1

2 − 12

and the nor-malization condition �+ 1

2 + 12+ �− 1

2 − 12

= 1, from Eq. (10) wecan express the steady-state nuclear spin polarization as

P = p+ − p−p+ + p− + κ

, (11)

where κ ≡ η/c(J ).As can be seen, the flipping probabilities p+ and p− play

an important role in this process. In the previous model [23],these were determined analytically from the simplified spinHamiltonian Eqs. (1)–(3) for the case S = 1 and I = 1/2 as

p+(B) = 2|〈0↓|+〉|2B |〈−1↑|+〉|2B , (12)

p−(B) = p+(−B), (13)

115206-4


where |+〉 is the eigenstate of HES(B). This state is a mixtureof |0↓〉 and |−1↑〉 states due to hyperfine coupling.

2. Our extended model

We now discuss the details of our extended model, whichis generally applicable and shows predictive power. First, weapply modifications to the spin Hamiltonian in Eq. (1).

In previous models, the external magnetic field was alignedparallel to the C3v axis of the defect. However, experimentsshow that DNP strongly depends on the misalignment of themagnetic field [23]. To describe the effect of this misalignment,we allow small deviations of the direction of the magneticfield from the symmetry axis of the defects. Furthermore,the nuclear Zeeman effect was also neglected in previousmodels. At stronger magnetic fields, such as at 500–1000 G,the nuclear Zeeman splitting can reach few megahertz, whichcan be comparable to the hyperfine interaction. We thus takethis effect into account in our model.

In the general case, the nucleus with nonzero spin is not onthe symmetry axis of the defect. In this case, the symmetry ofthe spin Hamiltonian is reduced, i.e., the hyperfine tensor Amay have three nondegenerate eigenvalues Axx , Ayy , and Azz,and the eigenvector az, which corresponds to the eigenvalueAzz, may have a nonzero angle of θ with the symmetry axis.The azimuthal angle ϕ may be chosen to zero without limitingthe generality. The effects of symmetry-breaking hyperfineinteractions have been included in previous considerations tosome extent [24,25]. However, a consistent description of theDNP process with a general hyperfine tensor has not beencarried out so far.

In our model, we consider nondiagonal hyperfine tensors,which can be parameterized by their eigenvalues and angle θ

as follows:

Hhyp = STAI = (US)TAdiag(UI), (14)

where U describes a rotation that transforms the Cartesianbasis to the eigenbasis of tensor A, and Adiag = UAUT is thediagonal tensor of elements Axx , Ayy , and Azz. Note that fora general hyperfine tensor the spin Hamiltonian may containS±Iz, SzI±, and S±I± terms that allow a wide range of spinrotation processes to occur (see Appendix for spin Hamiltonianmatrices).

The nuclear spin flipping probabilities play a key role inthe determination of the polarization, defined in Eq. (11). Themain innovation of our model is that it uses different definitionsfor the probabilities p+ and p− than previous models [23] andincludes important effects from the excited and ground states’spin Hamiltonians as well as external driving forces from thesurrounding spin bath. In the rest of this section, we describethe main concept of our new considerations.

For simplicity, in this section, we restrict ourselves tothe case of positive external magnetic field, which shifts theenergies of MS = +1 and −1 levels upward and downward,respectively. In such a case, at vicinity of BLAC, the MS = −1spin state mixes with the MS = 0 state. For simplicity, we donot consider the interaction of the MS = +1 state with otherstates here. However, we include it in our later calculations(see Sec. III B).

Nuclear spin rotation can occur both in the electronicground and excited states. The interplay of these rotations andnonradiative, spin-selective electronic decay is responsible forthe nuclear spin polarization. In the most general case, toachieve a net driving force toward nuclear spin polarization,the ground-state and the subsequent excited-state spin rotationprocesses have to fulfill the following criterion: starting fromthe MS = 0 electron spin state, the electron spin may beflipped into the MS = −1 state, but the starting nuclear spinstate must be flipped into the opposite spin projection. Whenelectron spin flip-flops occur, the MS = −1 electron spin isthen transported into the initial MS = 0 state by nonradiativedecay. Optical cycles can therefore flip nuclear spins. When therates of the flipping processes |0↓〉 → |0↑〉 and |0↑〉 → |0↓〉are different due to some sort of asymmetry, the repetitionof these dynamical cycles induces different population of thenuclear spin states and nonzero polarization.

All the possible spin rotation processes that fulfill theaforementioned criterion and are allowed by the general spinHamiltonian (see Appendix for details) are depicted on aschematic model of the dynamical cycle in Fig. 3. The prob-ability of the ground or excited state spin rotation processespSt(χInitial|χFinal), where χInitial and χFinal represents the initialand final spin configurations, respectively, can be determinedby the spin Hamiltonian of the defect in the considered states“St.” The probability of spin flip in a joint ground state-excitedstate spin rotation process is then the product of the proba-bilities of the two separate rotations. With the above require-ment, the products pGS(0,±1/2|χInter)pES(χInter|−1,∓1/2) andpGS(0,±1/2|χInter)pES(χInter|0,∓1/2) define the probability

0↓

-1↓

0↑

GS time evolution

-1↓

0↑

0↓

ES time evolution

Optical excitation

Decay

Final state Initial state 0↓ 0↑

-1↑

-1↓

0↑

0↓

GS time evolution

-1↑

-1↓

0↑

0↓

ES time evolution

Optical excitation

Decay

Final state Initial state

(a)

(b)

-1↓0↓0↑ 0↑

1. 3. 2.

4.

0↓

0↑

-1↑-1↑

0↓-1↑

FIG. 3. (Color online) A schematic diagram of the evolution of acoupled electron and nuclear spin system through the four steps of acomplete dynamic nuclear spin polarization cycle. The depicted spinrotation processes, shown by different paths along the black arrows,preserve the spin projection MS = 0 of the electron spin but flip thenuclear spin (a) from down to up and (b) from up to down, in a four-step mechanism. The states of up and down nuclear spin projectionsare depicted with blue and green backgrounds, respectively. In thegeneral case, the interplay of all of these parallel processes determinesthe net polarization of the nuclear spins, with the process rates definedby the ground and excited state spin Hamiltonians (see text andAppendix for more information). The upper and lower paths of arrowswith orange (thick light grey) outline show the most prominent spinrotation processes at the GSLAC and ESLAC for positive magneticfields, respectively. The background spin bath, which works againstDNP, is represented by grey background.

115206-5


of spin rotations that may induce driving forces toward nuclearspin polarization, where χInter represents an intermediate spinconfiguration and |±1/2〉 is a comprehensive notation for |↑〉or |↓〉 nuclear spin states.

In contrast to previous models, our two-state evaluationmodel requires suitable spin rotation mechanism for DNP notseparately but simultaneously in the ground and excited states.For example, at BGSLAC, the ground-state hyperfine couplingflips the nuclear and electron spins with high probability. Thisprocess is represented by orange arrows on the top panelsof Fig. 3. Optical excitation transports this spin state to theexcited state, where it starts to evolve in accordance with thenew spin Hamiltonian of the excited state. To obtain a highprobability for suitable electron and nuclear spin flips fromthe two successive time evaluations, the spin state must beunchanged in the excited state. For general hyperfine tensors ofthe excited and ground state that are not necessarily identical,this condition may not be the case. The role of minorityprocesses, represented with black arrows in Fig. 3, increases asthe symmetry of the system becomes more distorted. Examplesof this sort of distortion include a misaligned magnetic fieldor hyperfine interactions with low symmetry. Such effects canlower the polarizability of the system or cause unexpectedresonance effects, which can be captured by our model.

The next new consideration in our model is thedetermination of the preferential direction of the nuclear spinpolarization. The DNP process is in a stationary state when thepolarization and depolarization processes are in equilibrium[i.e., Eq. (10) is satisfied]. This state corresponds to acertain electron and nuclear spin configuration, which is notnecessarily equal to one of the energy eigenstates of the groundor excited-state spin Hamiltonian. Generally, the nuclear spinis polarized or preserved in a state that has the longest lifetimein the dynamical cycle. When the hyperfine interaction isisotropic or C3v symmetric, the energy eigenstate |0↑〉 has thelongest, quasi-infinite lifetime, since in this state there is nohyperfine coupling between the electronic and nuclear subsys-tems [23]. On the other hand, for general hyperfine interactionsof the excited and ground state, the longest-lived spin state,which is the stationary spin state, can be a nonsymmetricstate [28].

This nonsymmetric state can be written as a linear combi-nation |0〉 ⊗ (α|↑〉 + β|↓〉). If the nuclear spin state is theeigenstate of the nuclear spin operator Ie of quantizationaxis e, then unit vector e of the three-dimensional spaceshows the preferential direction for nuclear spin polarizationin the dynamical cycle. In our model, we determine this edirection for every different set of parameters. To find thisdirection, we used the following condition: the nuclear spinstate |↑e〉, which is the eigenstate of Ie with eigenvalue +1/2,has the longest lifetime when its precession is minimal. Theprecession would result in a continuous transition between thetwo eigenstates of Ie, i.e., |↑e〉 ↔ |↓e〉, which then reducesthe lifetime of the nuclear spin state and lowers polarization.Since the system spends most of the time in the ground stateduring the dynamical cycle, we assumed that the ground-stateprecession should be minimal in the stationary state of thedynamical cycle. This means that we minimize pGS(0,↑e|0,↓e)with respect to direction e. At the minimum, we find thedirection where the nuclear spin state |↑e〉 is preserved for

the longest time. This state is therefore the stationary nuclearspin state in the dynamical cycle.

In the optical cycle, the length of the free evolution ofthe spin system is limited by the lifetime and decoherenceof the electron states (see below). When the lifetime of theground or excited state is shorter than the characteristic timeof the spin rotation processes, the oscillatory probability isnot averaged out. In extreme cases, the spin-flip probabilityis effectively reduced by the short lifetime of the electronstate (see Fig. 8 for the case of the NV center in diamondin Sec. V A). This case is more pronounced in the excitedstate, since the excited state lifetime of both the NV centerin diamond and the divacancy in SiC is only 10–15 ns. Fora hyperfine interaction of 10 MHz, the oscillation time ofthe nuclear spin-flip probability is on the order of 100 ns atBLAC, is much longer than the lifetime of the excited state. Incontrast to previous models, we calculate the probabilities byexplicitly taking time into account in the spins’ excited-stateevolution (see Sec. III B for more detail). We assume that theevolution time in the excited state is exponentially decaying.The characteristic time of decay is connected to the lifetime ofthe excited state (τES) determined by experiment. On the otherhand, we assume that the oscillatory probabilities are averagedout in the ground state, where the system spends sufficient timefor this to happen.

Finally, we include additional important effects in ourmodel: electron spin dephasing and spin-lattice relaxationeffects. The surrounding spin bath of nuclear spins andparamagnetic defects disturbs not only the nuclear but theelectron spin of the considered spin system [28], causingdecoherence of the superposition states. This decoherence canbe described by the Lindblad equation,

∂ρ

∂t= − i

�[H ,ρ] + Lρ, (15)

where ρ and H are the density operator and the Hamiltonianof the considered system, respectively. The final term on theright-hand side of Eq. (15), which describes the effects of theenvironment, can be written as

Lρ =∑

n

γn

(CnρC†

n − 1

2{CnC

†n,ρ}

), (16)

where γn are the decay rates of processes described bythe Lindblad operators Cn. Decoupling of the superpositionstate of the |i〉 and |j 〉 states can be taken into account bythe operator C = |i〉〈i|−|j 〉〈j |. In our case, the state |+〉 =α|0↓〉 + β|−1↑〉, which is responsible for the most prominentnuclear spin flipping process, relaxes with T ∗

2 characteristictime due to the electron spin dephasing processes, can beaccounted by the Lindblad operator |0〉〈0|−|−1〉〈−1|. Besidesdephasing, electron spin-lattice relaxation can also hinder thenuclear spin flipping processes, by depopulating the hyperfinecoupled states |0↓〉 and |−1↑〉. This effect can be describedby the Lindblad operators |i〉〈j |. Recent experiments haveshown [55] that the electron spin coherence time T ∗

2 in theexcited state of the NV center in diamond is on the order of theexcited-state lifetime. To take into account the aforementionedeffects, we assume that the intact evolution time of the electronand nuclear spin system, described by the spin Hamiltonian in

115206-6


Eq. (1), is effectively reduced in the excited state due to theshort coherence time of the electron spin. Therefore, we used ascaled excited state lifetime τ ∗

ES = ντES in our model, where ν

is a free parameter of the property 0 < ν < 1. By consideringelectron spin decoherence and spin-lattice relaxation effects,1/τ ∗

ES ≈ 1/T ∗2 + 1/T1, where T1 is the characteristic time of

the spin-lattice relaxation. Since dephasing is the fastest spinrelaxation process, τ ∗

ES is assumed to be close to, but somewhatsmaller than T ∗

2 . For the ground state, where the evaluationtime is not included explicitly, we scaled down the probabilityof nuclear spin rotation processes by a factor of μ.

Using the results of experiment and first-principles cal-culations for the parameters of the above described model,three free parameters remain. All of them are related to someextent to the effect of decoherence and spin-lattice relaxation.Parameter κ ≡ η/c(J ) in Eq. (11) is mainly related to thenuclear spin-lattice relaxation, and the two parameters ν andμ, introduced above, are closely connected to the electrondephasing effects in the excited and ground states, respectively.

Despite these new considerations, the model that we havedescribed still has limitations. Since it is a single cycle model,i.e., evolution of the spin system is taken into account in asingle optical cycle, complicated processes that take place overmany cycles are not modeled. Such a mechanism appears forI � 1 [24,28].

B. Method of calculation of the nuclear spin polarization

In this section, we specify the equations that are used tocalculate the nuclear polarization P in the framework of ourmodel described in the previous section. As Eq. (11) shows, thepolarization can be calculated from the probability of nuclearspin up and down flips (p+ and p−) in a dynamical cycle,respectively, and from the rate of spontaneous nuclear spinflips κ due to the background spin bath. The latter quantity isone of our model’s free parameters. Since the LAC happens fortwo values of the external magnetic field, ±BLAC, the nuclearspin flipping probabilities can be divided up into two parts:

p+ = p(−1)+ + p

(+1)+ ,

(17)p− = p

(−1)− + p

(+1)− ,

where p(−1)+ and p

(−1)− and p

(+1)+ and p

(+1)− represent the

spin up and down flipping probabilities due to spin rotationprocesses in the subspace MS = {0,−1} and MS = {0,+1},respectively. For positive values of B, the second terms on theright-hand side of the equations have only a minor contribution.However, their role increases as B → 0. At B = 0, p+ andp− become equal, and therefore, P |B=0 = 0. In our model,we use the approximation [23] p

(+1)+ (B) = p

(−1)− (−B) and

p(+1)− = p

(−1)+ (−B). The nuclear spin-flipping probabilities,

corresponding to the MS = {0, − 1} subspace, can be definedas

p(−1)+ =

∑i

pGS(0↓|χi)[pES(χi |−1↑)� + pES(χi |0↑)],

p(−1)− =

∑i

pGS(0↑|χi)[pES(χi |−1↓)� + pES(χi |0↓)],

(18)

where � is the probability of nonradiative decay from theelectron spin state |±1〉 of the excited state to the ground-state spin state |0〉 and χi are the final and initial states ofthe ground and excited states’ time evolution, respectively.The summation goes over states |0↑〉,|0↓〉, |−1↑〉, and |−1↓〉,see Fig. 3.

To evaluate Eq. (18), we define the probabilitiespSt(χInitial|χFinal). For the ground-state spin rotation processes,we averaged out |〈χFinal|χ(t)〉|2 in time, as

pGS(χInitial|χFinal)

= σ GSI,F

1

TGS

∫ TGS

0|〈χFinal|e−iH GSt/�|χInitial〉|2dt , (19)

where we consider the integration time TGS → ∞. This limitmeans that the system spends enough time in the ground statefor the oscillatory probabilities to be completely averaged out.The factor σ GS

I,F takes into account the destructive effect of theelectron spin decoherence and spin-lattice relaxation. σ GS

I,F is atwo-value function that takes 1 when the initial and final statesare the same and takes 0 < μ < 1 when spin rotation occurs.The parameter μ is a fitting parameter of our model.

In contrast to that of the ground state, the excited state’slifetime is short. In this case, the flipping probabilities stronglydepend on the duration of the excited state’s evolution time,since it can be shorter than the periodicity of the oscillatoryprobabilities pES(χi |χf ), see Fig. 8. For a proper description,we have to include time in our considerations.

The flipping probabilities, correspond to the excited state’stime evolution, thus

pES(χInitial|χFinal) =∫ TES

0�(t)

∣∣〈χFinal|e−iH ESt/�|χInitial〉∣∣2

dt ,

(20)

where �(t) is the probability distribution function of theeffective length of the excited state’s evolution time. �(t) isassumed to be an exponential distribution

�(t) = 1

τ ∗ES

e−t/τ ∗ES , (21)

where τ ∗ES is the characteristic time of the decay. In our model,

τ ∗ES is the average effective time of evolution in the excited

state, which is considered to be proportional to the excitedstate’s lifetime τES,

τ ∗ES = ντES. (22)

Here, ν, which is the last free parameter of our model, is ascaling factor that takes into account electron spin relaxationeffects that can change the net evolution time.

Finally, as discussed during the description of the model,the preferential direction e of the nuclear spin polarization maydeviate from the C3v axis of the defect [28]. We determine thisdirection from the ground-state-spin Hamiltonian by findingthe direction e, where the nuclear spin state |↑e〉, which isan eigenstate of Ie with +1/2 eigenvalue, has the lowestprobability to flip into |↓e〉 in the ground state. This means thatthe precession of state |↑e〉 is minimal and this state is thereforethe stationary state of the dynamic nuclear spin polarization

115206-7


cycle. The criterion used is

mine

pGS(

0↑e

∣∣0↓e

). (23)

The calculation of the nuclear spin polarization P for agiven system defined by the spin Hamiltonian of Eq. (1) cannow be carried out by finding the suitable nuclear spin state|↑〉 and |↓〉 by Eq. (23), and using it to determine the flippingprobabilities of the spin state in accordance with Eqs. (19)and (20). From these values, the total probability of nuclearspin down-to-up and up-to-down flipping can be obtained byEq. (18), which provides the polarization of the nuclear spinvia Eq. (11).

The predefined parameters of the model are the diagonalelements and angles of the excited and ground state’s hyperfinetensors, AGS

xx , AGSyy , AGS

zz , θGS, AESxx , AES

yy , AESzz , and θES,

the zero-field-splitting parameter of the ground and excitedstate’s zero-field-splitting tensors, DGS and DES, the rate ofnonradiative decay �, and the excited state’s lifetime τES.The free parameters used to fit the theoretical curves to theexperimental ones are μ, ν, and κ .

IV. AB INITIO CALCULATIONS OF THE MODELPARAMETERS FOR THE NV CENTER IN DIAMOND

AND THE DIVACANCY IN 4H- AND 6H-SIC

Since some key parameters of the DNP model have notbeen measured, we apply ab initio methods to calculate them.In particular, we calculate the full hyperfine tensor of selectedproximate I = 1/2 isotopes in the ground and excited states.In addition, we identify the axial divacancy configurations in6H-SiC by calculating their DGS parameters in order to providea direct comparison for DNP processes in 6H-SiC.

We carry out first-principles density-functional theory(DFT) calculations to study the NV center in diamond andaxial divacancies in 4H- and 6H-SiC. We apply a 512-atomsupercell for diamond, and 576-atom and 432-atom supercellsfor 4H- and 6H-SiC, respectively. We apply �-point samplingof the Brillouin-zone, which suffices to ensure convergentcharge and spin densities. We utilize the plane-wave basisset together with the projector augmented wave method asimplemented in VASP5.3.5 code [56–59]. We apply our in-housecode to calculate the GS zero-field splitting from DFT wavefunctions [51,60], which has been tested and shown to providegood results with using Perdew-Burke-Ernzerhof (PBE) func-tional [61]. We apply the HSE06 hybrid functional [62,63]to calculate the hyperfine tensors of selected I = 1/2 nucleiwhere the spin polarization of the core electrons are takeninto account [42]. We apply the constrained DFT method tocalculate the ES spin density [43].

A. Results on NV center in diamond

The full hyperfine tensors of 15N and 13C coupled to NVcenters in diamond have not been experimentally determined.We thus apply ab initio calculations to obtain these parameters.The GS-hyperfine tensors of NV center in diamond havebeen recently characterized in detail [42]. In particular, weanalyze the hyperfine tensors in the GS and ES for coupled15N nuclei, for which the DNP was thoroughly studiedexperimentally [23]. We provided the ES hyperfine tensor

TABLE I. The calculated hyperfine tensors for the NV center indiamond in the ground (GS) and excited (ES) states. The hyperfineconstants (Axx , Ayy , Azz) are shown as well as the direction cosineof the largest Azz hyperfine constant represented by angle θ , whichis the angle between the direction of Azz and the symmetry axis.The Az hyperfine constant is the projected hyperfine tensor onto thesymmetry axis. The sites are defined by Smeltzer et al. [26].

Nucleussite, state Axx (MHz) Ayy (MHz) Azz (MHz) Az θ (◦)

15N, GS 3.9 3.9 3.4 3.8 015N, ES −38.5 −38.5 −58.1 −46.0 013Ca, GS 114.0 114.1 198.4 147.6 71.713Ca, ES 44.8 45.0 117.5 77.0 69.113CA, GS 12.7 12.8 18.5 14.9 72.013CA, ES 9.6 9.7 15.1 11.8 73.213CB, GS 11.5 11.6 17.0 13.6 68.313CB, ES 9.7 9.7 15.3 11.8 64.813CC, GS −10.3 −10.5 −8.4 −10.0 28.113CC, ES −6.9 −7.4 −3.6 −6.2 13.513CD, GS −6.8 −7.2 −3.8 −6.1 71.813CD, ES −7.4 −7.8 −5.3 −6.9 79.113CE, GS 2.9 3.0 4.8 3.6 29.313CE, ES 0.7 1.4 2.5 1.6 23.113CF, GS 4.5 4.9 2.9 4.2 55.113CF, ES 3.2 3.8 4.8 4.0 40.113CG, GS 2.1 2.2 3.5 2.6 75.313CG, ES 1.3 1.3 2.4 1.7 77.413CH, GS 1.0 1.0 2.0 1.4 15.013CH, ES 2.0 2.0 3.5 2.6 13.9

by PBE functional [25], where we showed that the hyperfineconstants are anisotropic for 15N. We list the correspondingHSE06 values in GS and ES in Table I.

B. Results on divacancies in 4H- and 6H-SiC

We calculate the electronic structure of the axial divacancydefects in 4H- and 6H-SiC in their neutral charge state withS = 1 electron spin. We have previously determined the DGS

parameter for hh and kk divacancies in 4H-SiC [51], and nowwe report it for hh, k1k1, and k2k2 divacancies in 6H-SiC. Welist the results in Table II. The results imply that QL1, QL2,and QL6 ODMR signals are associated with k1k1, hh, and k2k2

divacancies, respectively.Proximate 29Si nuclear spins of divacancies in GS have

been detected by means of ODMR [36], labeled as SiIIa andSiIIb by following the labels in Ref. [52] (see Fig. 2). Thehh and k2k2 divacancies show similar GS hyperfine constants,whereas the k1k1 divacancy exhibits larger values in 6H-SiC(see Table III). The k1k1 in 6H-SiC and kk in 4H-SiC as wellas hh configurations in 6H- and 4H-SiC show similar values.These trends support the assignment of QL1,2,6 based on thecalculated DGS parameters.

Finally, we calculated the corresponding hyperfine tensorsin the ES (see Table III). Within the accuracy of measurements,the ES shows signatures of C3v symmetry. However, a fullcharacterization of the divacancy’s ES is beyond the scopeof this study. We approximate the ES by the constraint DFTprocedure and fix the C3v symmetry in our calculations toobtain the ES hyperfine tensors.

115206-8


TABLE II. A summary of the spin-transition energies for the c-axis-oriented PL6 defect and neutral divacancies in 4H- and 6H-SiC.The parameters are all 20 K parameters, except for the DES of PL6,where the room-temperature value is given. Both DGS and DES arepositive. Comparing the experimental DGS with the calculated DGS

(calculated at T = 0 K, using the method in Refs. [51,60]) allowseach spin resonance transition in 6H-SiC to be corresponded with itsform of neutral divacancy.

Defect DGS (GHz) DcalcGS (GHz) DES (GHz)

4H: hh 1.336 1.358 0.844H: kk 1.305 1.320 0.784H: PL6 1.365 – 0.946H: hh 1.334 1.350 0.856H: k1k1 1.300 1.300 0.756H: k2k2 1.347 1.380 0.95

V. RESULTS AND DISCUSSION ON DYNAMIC SPINPOLARIZATION PROCESSES

In this section, we present our results and conclusions onthe DNP process for the NV center in diamond and for thedivacancy in 6H-SiC. First, we start with the study of the NVcenter in diamond, which is the most thoroughly investigateddefect system exhibiting DNP. We show that the new modelis capable of reproducing both theoretical and experimentalcurves. Nevertheless, it provides a new and deeper insightto the features of the DNP mechanism and to the physicsof the defect. After this, we apply our model for differentconfigurations of the divacancy in 6H-SiC. Our results revealthe importance of the electron spin coherence time in theexcited state in understanding DNP.

A. NV center in diamond and a single adjacent 15N nuclear spin

Previous models [23] successfully explained experimentalresults on the dynamic nuclear spin polarization of a 15Nnuclear spin of the NV center in diamond. In this specialcase, the hyperfine tensor of the excited and ground states aresymmetric (Table I). Therefore, when the magnetic field iswell aligned with the axis of the defect the spin Hamiltonianbecomes rather simple [23,25]. Here, we reassess theseexperimental and theoretical results in the framework of ourextended method.

To calculate the magnetic field dependence of the nuclearspin polarization, we chose parameters for our model asfollows. The parameters of the hyperfine tensor of the groundand excited state are determined by our first principlescalculations, Table I. The D parameters for the ground andexcited state zero-field-splitting tensors were set to 2.87 and1.42 GHz, respectively, in accordance with experiment. For theexcited state the lifetime and the rate of nonradiative decay,we used the experimental values τES = 12 ns and � = 0.3.The μ parameter in our model determines the features of theGSLAC resonance peak. Given the lack of experimental datafor this peak, we set μ = 0.25. The shape of the ESLACpolarization curve is determined by two parameters in ourmodel. Parameters κ and ν take into account the effect of thenuclear spin-lattice relaxation and the excited state’s electronspin decoherence, respectively. These parameters were fit tothe experimental curves P(B) and P(�B). The optimal valuesare ν = 0.13 and κ = 3.15 × 10−4.

In order to indicate the sensitivity of the fitted curve tothe variation of the free parameters, we depicted curves ofmodified ν and κ parameters in Fig. 5. As can be seen thetheoretical curve moderately but nonlinearly varies with thechanges of the fitting parameters.

We note that the fitting parameter κ has a smaller valuein our model than parameter k0

eq = 0.0027 of the previousmodel with same definition. The difference is due to the effectof the ES electron spin decoherence. In the previous model,this effect is not included, while in our model it is taken intoaccount explicitly. The electron spin decoherence shortens theeffective ES lifetime, thereby reducing the probabilities ofnuclear spin flips and the polarization. This finding immedi-ately draws attention to the importance of the electron spindecoherence.

Our extended model reproduces well the reported magneticfield dependence of the nuclear spin polarization [23] at thevicinity of BESLAC, see Fig. 4. The position of the maximumpolarization is at B = 516 G. It is slightly shifted due to thedifferent hyperfine tensors utilized in the two calculations.On the other hand, in our model, the ground-state processesare simultaneously described that produces an additionalresonance peak at BGSLAC = 1020 G. As the ground-statehyperfine interaction is weaker than the excited state inter-action, the GSLAC resonance peak is much sharper that at theESLAC.

TABLE III. The calculated hyperfine tensors for nuclear spins that are proximate to divacancies in 6H-SiC in the ground (GS) and exited(ES) state. The hyperfine constants (Axx , Ayy , Azz) are shown, and also the direction cosine θ , which is the angle between the direction ofAzz and the symmetry axis. The Az hyperfine constant is the projected hyperfine tensor onto the symmetry axis. The atom labels are shownin Fig. 2. Since PL6 in 4H-SiC has not yet been identified, we applied the calculated hyperfine tensors in the k2k2 divacancy configurationin the DNP simulations. In our experience, the difference in the measured or calculated hyperfine constants are within 1 MHz, and a 1 MHzinaccuracy does not alter the simulation results.

Nucleus Site Conf. St. Axx (MHz) Ayy (MHz) Azz (MHz) θ (deg.) Az (MHz)

29Si SiIIb 6H: hh GS 9.8 8.6 10.7 69.5 9.629Si SiIIb 6H: hh ES 9.8 9.3 10.4 63.0 9.829Si SiIIb 6H: k1k1 GS 10.7 9.8 11.5 70.8 10.529Si SiIIb 6H: k1k1 ES 10.2 9.5 10.8 60.7 10.129Si SiIIb 6H: k2k2 GS 9.9 8.7 10.8 69.4 9.729Si SiIIb 6H: k2k2 ES 10.4 9.7 11.0 64.1 10.3

115206-9


P

B [Gauss]

FIG. 4. (Color online) A comparison of calculated and measureddynamic nuclear spin polarization P of a 15N nucleus of the NV centerin diamond as a function of the external magnetic field B. The resultof our calculation is depicted with a thick red solid line, while theprevious theoretical and experimental results [23] are depicted withthin black solid line and black points, respectively. The calculatedESLAC resonance peak, at BESLAC = 516 G, resembles the reportedones [23]. However, in our calculation the ground-state processes arealso taken into account, producing an additional peak at BGSLAC =1020 G.

We also calculated the magnetic field angle dependenceof the degree of P , see Fig. 6 which was not considered byprevious models. The theoretical curve agrees well with theresult of the experimental measurements [23]. The polarizationdecays very quickly as the angle of the magnetic fieldincreases. Our model makes it possible to investigate theseobservations thoroughly. First of all, as the symmetry ofthe hyperfine tensor is reduced by the nonzero angle of themagnetic field and the symmetry axis of the defect in thestationary state, the nuclear spin is not parallel with either thesymmetry axis of the defect or the magnetic field. The angle θP

of the preferential direction of the polarization linearly dependson the angle θB of the magnetic field, see Fig. 6. The ratio ofthe two angles is θP/θB = 4.03.

P

B [Gauss]

(a)

B [Gauss]

(b)

P

+ 60%

+ 60%

FIG. 5. (Color online) Parameter dependence of the theoreticalnuclear spin polarization curve P(B) for the case NV center indiamond including an 15N nuclei. (a) and (b) show ν and κ parameterdependence of the theoretical curve, respectively. In both cases, thered thick curve corresponds to the optimal parameter setting, seeFig. 4, while the edges of the light purple filled area correspond to±60% change of the parameters. Variation of parameter μ has similareffect on the GSLAC peak as ν has on the ESLAC peak.

P

(a) (b)

FIG. 6. (Color online) The magnetic field angle dependence ofthe nuclear spin polarization. (a) shows the calculated (solid line) andmeasured [23] (points) spin polarization of a 15N nucleus of an NVcenter in diamond as a function of the angle of the external magneticfield and the C3v axis of the defect at B = 516 G. Our calculationaccurately reproduces the experimental observations. (b) The angle ofthe nuclear spin is depicted as function of the angle of the magneticfield. As the misaligned magnetic field reduces the symmetry, thestationary state of the nuclear spin points out of the C3v axis.

In Fig. 7, we depicted the magnetic field dependence of thenuclear spin polarization for the case of a misaligned magneticfield. The angle of the magnetic field and the symmetry axisof the defect is set to 1◦. The ESLAC resonance peak isstrongly reduced, whereas the GSLAC resonance peak almostdisappears.

Next, we investigate the role of different spin rotationprocesses when the symmetry is reduced. In Fig. 8, wecompare the time dependence of the ES spin flipping processesfor the case of aligned and misaligned magnetic fields. Inthe former case, only one spin rotation mechanism can beobserved, |0↓〉 → |−1↑〉, in accordance with the previousmodels [23,25]. By tilting the direction of the magnetic field,the maximal probability of this process decreases while otherflipping processes appear simultaneously with high probability[see Fig. 8(b)]. All of these mechanisms reduce the maximalpolarizability of the nuclear spin. For instance, |0↑〉 → |−1↓〉represents a driving force towards nuclear spin polarization

P

B [Gauss]

(a) BGSLAC

BESLAC

B [Gauss]

(b)

BESLAC BGSLAC

FIG. 7. (Color online) The calculated degree of dynamic nuclearspin polarization of the nucleus 15N of the NV center in diamondfor the case of misaligned magnetic field. The angle of the magneticfield and the C3v axis of the defect is set to 1◦. (a) The polarizationof the nuclear spin as a function of the strength of the magneticfield. It shows a strong reduction in the polarizability compared withFig. 4, due to the misalignment of magnetic field. Interestingly, theground-state dynamic nuclear spin polarization almost completelydisappears. (b) The angle of the nuclear spin and the C3v axis of thedefect as a function of the magnetic field.

115206-10

THEORETICAL MODEL OF DYNAMIC SPIN . . . PHYSICAL REVIEW B 92, 115206 (2015)p

Evolution time [ns]

(a)

Evolution time [ns] p

(b) -1 0 | |

-1 0 | | 0 | 0 |

-1 0 | |

ES

ES *

FIG. 8. (Color online) The time dependence of spin flippingprobabilities in the excited state of the NV center in diamond with a15N nucleus. (a) The magnetic field is well aligned with the axis of thedefect. In this case, only one spin rotation can occur, |0↓〉 → |−1↑〉,represented with red (thick gray) line. The black and blue lines withlight gray and blue filled areas represent the exponential decay of theexcited state lifetime and the effective evolution time, respectively(see text for more explanation). The mean evolution time can be seento be much shorter than the periodicity of the oscillatory probability.(b) The magnetic field is misaligned by 1◦. In this case, different kindsof spin rotations occur that lower the polarizability of the nuclear spin(not all processes are depicted).

in the opposite direction, while |0↓〉 → |0↑〉 represents theprecession of the nuclear spin that reduces the polarizationregardless the direction of the nuclear spin. The interplayof these effects is responsible for the reduction of the ESpolarizability.

From Fig. 8, it is clear that the ES lifetime and the lengthof intact evolution time play an important role in DNP. Byfitting the theoretical curve to the experimental data P(�B),we obtain the average length of the intact evolution time,which turns out to be only 14% of the ES lifetime. Thisshort time suggests substantial electron spin decoherence inthe excited state. In such circumstances, the spin rotationprocesses have a longer time scale than the net evolution time.There are two main consequences of this. First, the nuclearspin flipping probabilities are largely reduced, since the spinrotations are suppressed. Second, there is a preference over fastprocesses. For example, in the case of misaligned magneticfield, the fastest mechanism, exhibiting rapid oscillations, canflip the spins with the highest probability during the short timeevolution. Thus, at the ESLAC, the |0↓〉 → |−1↑〉 processis still dominant causing nonzero nuclear spin polarization.On the other hand, by elongating the ES evolution time, thispreference relaxes and other slower processes can have morepronounced effects [see Fig. 8(b)], which would greatly changethe nuclear spin polarization pattern. This can be the reasonfor the disappearance of the nuclear spin polarization at theGSLAC (see Fig. 7), where the ground state’s evolution timeis much longer than that of the excited state.

Finally, we emphasize the strong relation between the mag-netic field angle dependence of the nuclear spin polarizationand the electron spin decoherence in the ES. Shorter coherencetime reduces the maximal polarizability but the decay of thepolarization with respect to angle �B of the magnetic field isslower.

B. NV center in diamond and first neighbor 13C nuclear spin

In this section, we study the dynamic nuclear spin polariza-tion of a 13C nuclei in the first neighbor shell, Ca site, of theNV center, where both the ESLAC and GSLAC polarizationwas observed [23,39]. In the latter case, particular featuresappear due to the nonsymmetric hyperfine interaction, whichshow a potential for new applications in the area of sensitivity-enhanced nuclear magnetic resonance and spintronics [39].

As a N atom with nonzero nuclear spin is an inherent partof the NV center, the inclusion of an adjacent 13C nucleusnecessarily results in a three-spin system. Due to the smallgyromagnetic ratio of the nuclei, the direct nuclear spin-nuclear spin interaction is negligible. Substantial interaction is,however, mediated by the electron. In this case, the hyperfineinteraction with the nitrogen atom appears as an additionalmagnetic field dependent electron spin relaxation effect forthe two-spin system of the nucleus 13C and the electron. Onthe other hand, in the stationary state of the system, the nitrogennuclear spin is highly polarized, while the populated spin stateof the nitrogen nucleus is only weakly coupled to the electronspin. Therefore we neglect the effect of the nitrogen nuclearspin on the polarizability of the 13C nucleus. We investigatethe system as a two-spin system within our model. Descriptionof three-spin DNP systems can be found in Ref. [44].

In the calculation, we used the hyperfine parameterspresented in Table I. Other parameters are the same as inthe previous section. The reason for keeping the parametersis that the measurements on the ESLAC polarizability of 14Nand 13C nuclei were carried out on the same sample [23].

As can be seen in Fig. 9, the predicted maximal nuclearspin polarization of 70% at ESLAC is slightly below theexperimentally observed 90% [23]. Since the measurementswere carried out on single centers, the local environmentis not necessarily the same, which may be the reason forthe differences. By assuming weaker spin dephasing effects,higher polarization can be achieved in the calculations.

The oscillatory features of the calculated nuclear spinpolarization of the 13C nucleus at GSLAC [see Fig. 9(a)]reproduces well the experimental observations by Wanget al. [39]. In our case the nuclear spin polarization reaches83% at 950 G before it rapidly drops at GSLAC. Right after

P

B [Gauss]

(a) BGSLAC

BESLAC

B [Gauss]

(b)

BESLAC BGSLAC

FIG. 9. (Color online) Calculated magnetic field dependence ofthe dynamic nuclear spin polarization of a single nucleus 13C in thefirst neighbor shell of the NV center in diamond. (a) The polarizationof the nuclear spin as a function of the strength of the magnetic field.(b) The angle of the nuclear spin and the C3v axis of the defect as afunction of the magnetic field.

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the GSLAC, the polarization reverses and reaches −11% at1050 G. It then increases again and reaches 62% at 1170 G.This feature has been thoroughly investigated and understoodby Wang et al. [39]. In short, due to nonsymmetric hyperfineinteractions, unusual combinations of ladder operators appearin the spin Hamiltonian (see Appendix). From Figs. 3(a)and 3(b), it is clear that the reverse of the polarization isdue to the dominance of process |0↑〉 → |−1↓〉 over themost common process |0↓〉 → |−1↑〉 in the ground state.This dominant process occurring only at well-defined externalmagnetic fields is governed by the S−I− operator combinationthat resonantly mixes state |0↑〉 and |−1↓〉 at a certainmagnetic field slightly higher than BGSLAC. The resonanceappears as a dip due to the inverse polarization. Since thehyperfine constant correspond to term S−I− is smaller thanthe one correspond to the regular term S−I+, the dip is sharperthan the resonance peak due to the term S−I+. In summary,GSLAC polarization curve built up from a wide resonancepeak, due to term S−I+, and a slightly shifted sharper dip, dueto term S−I−.

By looking at the polarization near the ESLAC (Fig. 9),one may recognize similar oscillation of the polarization.However, here the pattern is not as developed as for theGSLAC resonance. From the discussion presented in theprevious section, it is clear that the short ES lifetime andthe fast electron spin dephasing effect cause the difference.The hyperfine constant of term S−I+ is larger than that ofS−I− [39], which implies slower spin rotation processes forthe latter term. As the short evolution time reduces the flippingcapability of slow processes, the dip corresponding to theslower S−I− process is effectively suppressed. By manuallyincreasing the ES lifetime, we can see a more developed dip.

Finally, the magnetic field dependence of the polarizationdirection of the 13C nucleus exhibits similar pattern at theGSLAC [see Fig. 9(b)] as 15N for the case of misalignedmagnetic field. One can argue that the two types of symmetryreduction of the spin Hamiltonian result in qualitatively thesame phenomena. This realization may provide a way toengineer the magnetic field dependence of the polarizationof nonsymmetrically placed nuclei.

C. NV center in diamond and remote 13C nuclear spins

The study of the polarizability of remote 13C nuclearspins provides an opportunity to have a deep insight into theparameter dependence of the maximal achievable nuclear spinpolarization in DNP processes, which is thoroughly examinedin the first section. Our conclusions are tested on the availableexperimental results in the second subsection.

1. Parameter dependence of nuclear spin polarization

From the applications point of view, it is important tounderstand the degree to which the spin polarization of theelectrons can be transferred to the nuclei by DNP and whatthe limiting factors are. Remote nuclear spins couple weaklyto the electron spin of the defect, and thus their polarizabilityis assumed to strongly depend on the details of the hyperfinetensor. In this section, we aim to understand the role of differenthyperfine parameters in the determination of the maximalachievable nuclear spin polarization of more remote nuclei that

couples relatively weakly to the electron spin of an NV center.Previous publications have already addressed this issue to anextent [24–27]. However, a satisfactory conclusion has notbeen achieved so far. The most commonly mentioned reasonsfor the varying polarizability of remote 13C nuclei are thevarying angle of the principal axis of the ES hyperfine tensorand the symmetry axis of the defect as well as the varying angleof the principal axis of the ES and GS hyperfine tensor. On theother hand, as we will show, the strength of the coupling mayhave the most important role.

In this section, we investigate a set of 13C (I = 1/2) nuclearspins interacting with the electron spin of the NV center indiamond at ESLAC condition. We examined the achievablenuclear spin polarization as a function of the strength ofthe coupling, the anisotropy of the hyperfine interaction, thedirection cosines of the ES hyperfine tensor, and the differenceof the direction cosines of the ES and GS hyperfine tensors.In the calculations, we arbitrarily vary AES

xx = AESyy = AES

‖ , andAES

zz = AES⊥ in the range of 1–24 MHz and θES and θGS in the

range of 0–90◦ to map the parameter space, while we keptfixed other parameters of the model. The free parameters wereinitialized as κ = 0.0015, ν = 0.5, and μ = 1.

First, we investigate the effect of GS processes on theESLAC polarizability of weakly coupled nuclear spins. TheGS-spin Hamiltonian plays an important role, since the systemspends most of the time in GS. In our model, this Hamiltoniandetermines the direction of nuclear spin polarization. Thissuggests that the details of the GS hyperfine interaction canaffect the polarizability. Our calculations, however, show thatthe nuclear spin polarization direction is parallel with the axisof the defect, i.e., the nuclear spin polarized parallel to theelectron spin, regardless the details of the GS hyperfine tensor.Thus we can say that the difference of the principal axis ofthe GS and ES hyperfine tensor, which is thought to play animportant role, as well as other parameters of the GS hyperfinetensor, do not affect the ESLAC polarizability of the spinof remote nuclei. One may prove this observation based onsimple analytical arguments. Consider the usual basis |0↑〉,|0↓〉, |−1↑〉, and |−1↓〉, where the quantization axis of theelectron and nuclear spins are parallel to the symmetry axis ofthe defect. Far from the GSLAC, the large zero field splittingof the electron spin states suppresses the mixing of the states,while at the GSLAC, the mixing is more enhanced due tothe GS hyperfine coupling. On the other hand, for the caseof weak GS interactions, the mixing and other effects of theinteraction are simply negligible at the ESLAC, since A ≈O(10 MHz) � 1400 MHz ≈ DGS − DES. Thus, the state |0↑〉and |0↓〉 are eigenstates of the GS spin Hamiltonian withingood approximation. Their long GS lifetime makes them goodtarget states for DNP polarization.

To investigate the dependence of the polarizability on thedetails of the ES hyperfine tensor, we depict the maximalpolarization of the 13C nuclear spin as a function of theprincipal axis of the hyperfine tensor for different couplingstrength A⊥ in Fig. 10(a). As one can observe, the angulardependence of the polarization is not substantial in mostcases. However, it depends on the anisotropy of the hyperfineinteraction. For example, when the interaction is isotropic,i.e., A⊥ = A‖, no angular dependence is clearly observed.However, when the interaction is highly anisotropic more

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FIG. 10. (Color online) Parameter dependence of the dynamicnuclear spin polarization of 13C nuclei adjacent to a NV center indiamond. (a) The polarization is plotted against the angle of C3v

axis and the principal axis of the ES hyperfine tensor for differentvalues of coupling strength A⊥. (b) The polarization is plotted againstthe coupling strength A⊥. The hyperfine parameters are given inmegahertz and refer to ES interactions.

pronounced angular dependence can be observed, see the caseof A⊥/A‖ = 1/8 or A⊥/A‖ = 3. To understand the observedfeatures, we need to take into account three effects: First, thevalue of the off-diagonal elements, which are responsible forthe spin flipping processes, are functions of angle θ as

〈−1↑|Hhyp|0↓〉 =√

2

4(A⊥(1 + cos2(θ )) + A‖ sin2(θ )).

(24)

For instance, the mixing term increases with angle θ asA‖ > A⊥, which results in increasing polarizability for smallθ , see the case of A⊥ = 1 and 2 MHz in Fig. 10(a). Theopposite effect can be observed for A‖ < A⊥. This effectincreases with the anisotropy of the hyperfine tensor. Second,as the angle θ increases, the symmetry of the spin Hamiltonianbecomes more and more reduced, thereby strengthening thenuclear spin depolarization processes [see Fig. 8(b)]. Third,the effect of these processes depends on the net evolutionlimited by the lifetime of the excited state and the electronspin coherence time in the excited state. The interplay ofthe above described two basic mechanisms determines thehyperfine angle dependence of the polarizability.

Next, we investigate DNP as a function of the strength ofthe hyperfine coupling. We depict the maximal polarizationagainst parameter A⊥ in Fig. 10(b) with fixed A‖/A⊥ ratioand angle θ . As one can see, the polarization quickly drops asthe hyperfine interaction decreases below 10 MHz. The reasonof the reduction of the polarizability is the finite evolutiontime in the ES of the dynamical cycle. As the strength ofthe coupling decreases, the rate of the spin flipping processdecreases as well. Due to the finite lifetime, the time is notsufficient to flip the spins. On the other hand, the characterof the curve depicted on Fig. 10(b) strongly depends onthe length of the effective evolution time and thus on thestrength of the electron spin dephasing effects. We can saythat the finite lifetime and the short electron spin coherencetime of the ES limits DNP of remote nuclear spins. Shorterevolution times cause the drop of the polarizability at largerhyperfine couplings. Here, we mention that the enrichmentof the nuclear spin concentration effectively decreases the

electron spin coherence time, thus suppressing DNP processesof remote and nearby nuclear spins. This mechanism canhave important consequences on the NMR applications ofthese DNP processes, and may suggest the importance ofimplementing dynamical decoupling protocols designed toenhance DNP.

To summarize our findings on the hyperfine parameterdependence of the DNP process, we can say that, at a givenexternal condition, the polarizability of remote 13C nuclearspins is determined by their A⊥ parameter in the first order. Adiverse secondary contribution is governed by the relationshipof the three parameters A⊥, A‖, and θ . The details of theGS hyperfine interaction do not affect the polarizability. Andfinally, the electron spin coherence time limits how far theelectron spin polarization can be transferred by ESLAC DNPmechanism.

2. Comparison of theory and experiment for the caseof remote 13C nuclei

In this section, we compare the results of our calculationswith the available experimental measurements for the caseof distant 13C nuclei around an NV center in diamond (seeFig. 11). In our calculations, we used the calculated hyperfinecoupling parameters listed in Table I, while the zero-field-splitting parameters, the ES lifetime, and the nonradiativedecay rate were set to be the same as in the previous sections.The free parameters were specified as κ = 0.003, μ = 0.9,and ν = 1. In the calculations, the magnetic field was set toB = 510 G as given in the experiments.

FIG. 11. (Color online) The degree of dynamic nuclear spinpolarization of 13C nuclei at various sites around a NV center indiamond. The calculated polarizations (the red line with stars) arecompared with the available experimental results. Blue squares andgreen triangles show the experimental results by Smeltzer et al. [26]and Dreau et al. [27], respectively. The sign of the experimentalpolarization is changed for sites C and D, see text for moreexplanation. Sites E and F and sites G and H could not be separated inthe experiment due to their similar GS hyperfine parameters. Thus themeasured polarization can be the mixture of the polarization of thetwo sites. As sites F and H with threefold degeneracy are more scarcethan the corresponding E and G sites with sixfold degeneracy, F andH data points are represented with empty symbols. Because only afew centers with 13C nuclei on a specific site were investigated, it ispossible that only the most probable sites, E and G, were measured.

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The calculated polarization is in good agreement with theexperimental measurements by Smeltzer et al. [26]. However,the agreement is worse for the ones obtained by Dreauet al. [27] (see Fig. 11). One should keep in mind thatthere are many factors that may limit the direct comparisonbetween theory and experiments. Beside the experimentaluncertainties, the calculated ES hyperfine parameters havesystematic uncertainties due to the approximate theory and thedifficulties of the description the ES-electron configuration.

By looking at the theoretical result in Fig. 11 as well asat the hyperfine parameters in Table I, the basic roles forthe polarizability, defined in the previous subsection, can berecognized. However, there is an additional factor one shouldtake into account. Different 13C sites exhibit different hyperfinecoupling to the electron spin of the NV center. By applyingthe same magnetic field for all the different sites the resonancecondition is not satisfied equally for all the nuclei spins. As thehyperfine coupling is relatively weak and the resonance peaksare sharp, the polarization pattern among the different sites cansharply depend on the value of the external magnetic field. Thismay be a reason for the substantial deviation between the twoavailable experimental measurements. In our calculations, wefound that the variation of �B = ±5 G can result in 5%–25%variation of the observed polarization among the different sites.Our previously defined rules apply to maximally achievablepolarization and thus to perfect resonance conditions. Theimperfect resonance condition overshadows the trends dictatedby our findings.

Our calculations show positive polarization for both posi-tive and negative hyperfine parameters. However, in previousmanuscripts, negative hyperfine parameters were associatedwith negative polarization [26,27]. We think that the dis-crepancy comes from a misinterpretation of the sign ofthe polarization in the experiments. For positive hyperfineparameters, the population of the lower energy eigenstaterepresents positive nuclear spin polarization. On the otherhand, for negative hyperfine parameters, positive polarizationcorresponds to the population of the higher energy eigenstates.In the measurements, the population of the higher energyeigenstate is interpreted as negative polarization. However,it is actually due to the change of the sign of the hyperfineparameter. Thus, in Fig. 11, we depict the experimentalpolarization of C and D sites with positive sign, in agreementwith our simulations.

D. Divacancy in 6H-SiC and adjacent 29Si nuclear spin

Recently, dynamic nuclear spin polarization of the hh andk1k1 configurations of the divacancy in 6H-SiC and the relatedPL6 center in 4H-SiC were observed [36]. In this section,we use our model to reproduce the measured magnetic fielddependence of the nuclear spin polarization at the vicinityof the ESLAC and to use the fitting parameters to gainimportant insight to the DNP processes. Here, we restrictourselves to study of ESLAC DNP for the different divacancyconfigurations.

Considering the experimental results [36] and the calculatedhyperfine parameters of the different configurations, seeTable III, an interesting observation can be made. Although,the hyperfine parameters of these configurations differ by

P

B [Gauss]

QL1 QL2 PL6

FIG. 12. (Color online) The adjusted theoretical curves (solidlines) and the experimental measurements [36] (points) of the dy-namic nuclear spin polarization of different divacancy configurationsand related defects. The panels show the QL1, QL2, and PL6photoluminescence centers (ordered from left to right). The first twocenters are assigned to the k1k1 and hh configurations of divacancyin 6H-SiC (see text for further information). The unidentified PL6center is modeled by hyperfine parameters of k2k2 configuration inthe theoretical simulations.

only a few MHz, the measured nuclear spin polarization varybetween 60% and 99% at ESLAC, see Fig. 12. As we will showin this section, such large variation of the polarizability canonly be explained by the different ES electron spin coherencetimes of the different configurations. This observation furtheremphasizes the role of these effects in the DNP process.

The hyperfine parameters, used in the DNP calcula-tions, were determined by our ab initio calculations, seeSection IV. Recent experiments revealed sensitive temperaturedependence of the excited state zero-field-splitting of thedivacancy [36]. Since the DNP measurements were carried outat various temperatures [36], we also adjusted DES in our DNPsimulations. The EES parameter is set to zero in all cases. TheD parameter together with the measured rate of nonradiativedecay � and ES lifetime τES are listed in Table IV. The fittingparameters correspond to the different type of spin relaxationprocesses are collected in Table V.

The adjusted theoretical curves and the experimentalmeasurements are depicted together in Fig. 12 for hh andk1k1 configuration of divacancy in 6H-SiC and PL6 centerin 4H-SiC. Good agreement is achieved for all the cases.We would like to emphasize that the theoretical curves areadjusted by the variation of parameters that are related to spin

TABLE IV. The utilized excited state zero-field-splitting param-eter DES as well as the measured [51] rate of nonradiative decay �

and excited state lifetime τES are listed for hh and k1k1 configurationof divacancy in 6H-SiC and the related PL6 defect in 4H-SiC.

Configuration, SiC polytype DES (MHz) � τES (ns)

hh, 6H 930 0.15 15k1k1, 6H 915 0.16 15PL6, 4H 955 0.33 14

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TABLE V. Fitting parameters κ and ν for hh and k1k1 configura-tion of divacancies in 6H-SiC and the related PL6 defect in 4H-SiC.The latter parameter determines the effective evolution time τ ∗

ES whichis then compared with the experimentally measured [36] coherencetime T ∗

2 . In addition, the ambient measurement temperature T isgiven.

Configuration,SiC polytype κ ν τ ∗

ES = ντES (ns) T ∗2 (ns) T (K)

hh, 6H 8.25 × 10−5 0.092 1.38 4.0 100k1k1, 6H 8.25 × 10−5 0.045 0.68 1.3 100PL6, 4H 3.14 × 10−4 0.362 5.1 5.0 300

decoherence and spin-lattice relaxation processes. Throughthe effect of the variation of these parameters, the effectof different spin relaxation processes may be understood.The κ parameter, corresponding to the rate of spontaneousnuclear spin flipping and thus to the nuclear spin-relaxationtime, primarily determines the decay of the polarizationcurve away from the ESLAC resonance. Second, it lowersthe maximal achievable polarization at resonance. The ν

parameter corresponds to the effect of electron decoherencein the ES. It mainly determines the height of the resonancepeak and thus the maximal polarization at ESLAC.

Spin relaxation processes are taken into account ratherapproximately in our model, which might describe either theintrinsic properties of the defects or the different environmentof the defects. Considering the fitting parameters in Table V,one can see that κ has larger value for PL6 in 4H-SiCthan for hh and k1k1 in 6H-SiC, suggesting faster nuclearspin-lattice relaxation for PL6. The reason can be threefold:(i) the measurements on the PL6 center were carried out atroom temperature while in the other cases at 100 K; (ii)the 4H sample is processed differently than the 6H samplewith possibly creating different number of disturbing spins thecrystals; (iii) PL6 is preferentially located relatively close to thesurface [15] where the concentration of noncoherent electronspins may be larger. In Table V, one also finds the values ofthe fitting parameter ν, together with the estimated effectiveevolution time τ ∗

ES with the trend of k1k1 < hh < PL6. Thistrend is good agreement with the one that can be observedfrom the measured electron dephasing time T ∗

2 , see Table V,governing the length of the effective evolution time in theexcited state τ ∗

ES in our model. This result indicates that theachievable high (low) ESLAC polarizability for PL6 (k1k1)is due to the longer (shorter) electron coherence time inthe ES.

VI. SUMMARY

We developed a model that provides a detailed descriptionof the dynamic nuclear spin polarization process of pointdefects with a high ground-state electronic spin and adjacentnuclear spins. It takes into account many microscopic featuresand processes that previous models have overlooked or notintegrated: the effect of the electron decoherence in theground and excited state, the short lifetime of the excitedstate, the direction of the nuclear spin polarization, theanisotropy of hyperfine tensors, misaligned magnetic fields,

and the simultaneous flipping of ground and excited state’sspin flipping processes. We applied our model to differentcases of the NV center in diamond and divacancies andrelated defects in 4H- and 6H-SiC, validating our model andshowing its generality. Additionally, in many cases, the modelprovided a new insight to the microscopic mechanisms behindthe examined phenomena and allowed us to draw importantconclusions. We show that the electron decoherence in the ESplays an important role in DNP process, through the shorteningthe ES evolution time. It can limit the maximal achievable ESnuclear spin polarization, as we showed for the divacancyin SiC. By promoting fast nuclear spin rotation processes andsuppressing slower ones, the electron decoherence protects theESLAC resonance from secondary spin rotation mechanismsthat appear for anisotropic hyperfine interactions. On the otherhand, similar secondary processes produce more complicatedfine structure at GSLAC, due to a longer evolution timepermitted by the longer lifetime of the GS in the optical cycle.By studying the polarizability of remote nuclear spins, weaddressed the long-standing question concerning the hyperfineparameter dependence of the maximal achievable polarization.As we showed, the strength of A⊥ primarily governs thepolarizability of remote nuclei, while the ES electron spindecoherence is an important factor as well. Throughout ourinvestigations, electron spin decoherence appeared as animportant, sometimes major, limitation of the polarizabilityof the nuclear spins. Therefore, any treatment that affectsthe point defect electron spin coherence times, like isotopeenrichment, could drastically affect the DNP process aswell.

ACKNOWLEDGMENTS

Discussions with Zoltan Bodrog are highly appreciated.Support from the Knut & Alice Wallenberg Foundation “Iso-topic Control for Ultimate Materials Properties,” the SwedishResearch Council (VR) Grants No. 621-2011-4426 and 621-2011-4249, the Swedish Foundation for Strategic Researchprogram SRL Grant No. 10-0026, the Grant of Ministryof Education and Science of the Russian Federation (GrantNo. 14.Y26.31.0005), the Tomsk State University AcademicD. I. Mendeleev Fund Program (project No. 8.1.18.2015), theSwedish National Infrastructure for Computing Grants No.SNIC 2013/1-331, and the “Lendulet program” of HungarianAcademy of Sciences is acknowledged.

APPENDIX: SPIN HAMILTONIAN MATRICES

Here, we specify the spin Hamiltonian matrices for thecoupled system of a S = 1 electron spin and an I = 1/2nuclear spin. The spin Hamiltonian consists of four terms [seeEq. (1) in the main text]. The corresponding four matrices, aswritten in the basis of |+1↑〉, |+1↓〉, |0↑〉, |0↓〉, |−1↑〉, and|−1↓〉 states, where the spin quantization direction z is parallelto the C3 axis of the defect, are listed below. The occurrenceof nonzero off-diagonal elements, which are responsible forspin rotation processes among the basis states, are discussedin each case.

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(1) The zero-field interaction term

Hzfs(D,E) =

⎛⎜⎜⎜⎜⎜⎝

D 0 0 0 E 00 D 0 0 0 E

0 0 0 0 0 00 0 0 0 0 0E 0 0 0 D 00 E 0 0 0 D

⎞⎟⎟⎟⎟⎟⎠

, (A1)

where D and E are the zero-field-splitting parameters. For point defects of C3v symmetry, the E parameter is zero. Thus thereare no off-diagonal elements in the spin Hamiltonian due to the zero-field interaction.

(2) Electron spin Zeeman term

HZ(B,θB) = geμBB

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

cos θB 0 1√2

sin θB 0 0 0

0 cos θB 0 1√2

sin θB 0 01√2

sin θB 0 0 0 1√2

sin θB 0

0 1√2

sin θB 0 0 0 1√2

sin θB

0 0 1√2

sin θB 0 − cos θB 0

0 0 0 1√2

sin θB 0 − cos θB

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

, (A2)

where B is the strength of the external magnetic field, θB is angle of the C3 axis of the defect and the external magnetic field, ge

is the g factor of the electron spin, and μB is the Bohr magneton.As can been seen, off-diagonal matrix elements appear for θB �= 0. These elements correspond to the electron spin ladder

operators S+ and S− and couple |MS ,MI 〉 and |MS±1,MI 〉 states.(3) Nuclear spin Zeeman term

HNZ(B,θB) = 1

2gNμNB

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

cos θB sin θB 0 0 0 0

sin θB − cos θB 0 0 0 0

0 0 cos θB sin θB 0 0

0 0 sin θB − cos θB 0 0

0 0 0 0 cos θB sin θB

0 0 0 0 sin θB − cos θB

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

, (A3)

where B is the strength of the external magnetic field, θB is angle of the C3 axis of the defect and the external magnetic field, gN

is the g factor of the nuclear spin, and μN is the nuclear magneton.Similarly to the electron spin Zeeman term, off-diagonal matrix elements are present for θB �= 0. Here, these off-diagonal

elements correspond to the nuclear spin ladder operators I+ and I− and couple |MS ,MI 〉 and |MS ,MI±1〉 states.(4) Hyperfine interaction term

Hhyp(Axx,Ayy,Azz,θ

) = 1

2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

a b 1√2b 1√

2c− 0 0

b −a 1√2c+ − 1√

2b 0 0

1√2b 1√

2c+ 0 0 1√

2b 1√

2c−

1√2c− − 1√

2b 0 0 1√

2c+ − 1√

2b

0 0 1√2b 1√

2c+ −a −b

0 0 1√2c− − 1√

2b −b a

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, (A4)

where

a = Azz cos2 θ + Axx sin2 θ, (A5)

b = (Azz − Axx) cos θ sin θ, (A6)

c± = Axx cos2 θ + Azz sin2 θ±Ayy , (A7)

where Axx , Ayy , and Azz define the hyperfine coupling strength and θ defines the polar angle of the principal axis of the hyperfinetensor.

For the case of θ = 0 and Axx = Ayy = A⊥, the nonzero off diagonal matrix elements 12√

2c+ = 1√

2A⊥ correspond to the S±I∓

operator combinations. These elements mix |MS ,MI 〉 and |MS±1,MI∓1〉 states. When Axx �= Ayy but θ = 0, new off-diagonal

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elements are present in connection to the S±I± operators. The coupling strength is 12√

2c− = 1

2√

2(Axx − Ayy) and the coupled

states are |MS ,MI 〉 and |MS±1,MI±1〉. When θ �= 0 and θ �= nπ2 , where n is an integer, precession like spin rotation process can

appear due to the S±Iz and SzI± operators that couple state similar to the electron and nuclear spin Zeeman effects, respectively.

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